#region Using

using System;
using System.Runtime.Serialization;

#endregion

namespace DotNetMatrix {
	/// <summary>Singular Value Decomposition.
	/// <P>
	/// For an m-by-n matrix A with m >= n, the singular value decomposition is
	/// an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and
	/// an n-by-n orthogonal matrix V so that A = U*S*V'.
	/// <P>
	/// The singular values, sigma[k] = S[k][k], are ordered so that
	/// sigma[0] >= sigma[1] >= ... >= sigma[n-1].
	/// <P>
	/// The singular value decompostion always exists, so the constructor will
	/// never fail.  The matrix condition number and the effective numerical
	/// rank can be computed from this decomposition.
	/// </summary>
	[Serializable]
	public class SingularValueDecomposition : ISerializable {
		#region Class variables

		/// <summary>Arrays for internal storage of U and V.
		/// @serial internal storage of U.
		/// @serial internal storage of V.
		/// </summary>
		private double[][] U, V;

		/// <summary>Row and column dimensions.
		/// @serial row dimension.
		/// @serial column dimension.
		/// </summary>
		private int m, n;

		/// <summary>Array for internal storage of singular values.
		/// @serial internal storage of singular values.
		/// </summary>
		private double[] s;

		#endregion   //Class variables

		#region Constructor

		/// <summary>Construct the singular value decomposition</summary>
		/// <param name="Arg">   Rectangular matrix
		/// </param>
		/// <returns>     Structure to access U, S and V.
		/// </returns>
		public SingularValueDecomposition(Matrix Arg) {
			// Derived from LINPACK code.
			// Initialize.
			var A = Arg.ArrayCopy;
			m = Arg.RowDimension;
			n = Arg.ColumnDimension;
			int nu = Math.Min(m, n);
			s = new double[Math.Min(m + 1, n)];
			U = new double[m][];
			for(int i = 0; i < m; i++) {
				U[i] = new double[nu];
			}
			V = new double[n][];
			for(int i2 = 0; i2 < n; i2++) {
				V[i2] = new double[n];
			}
			var e = new double[n];
			var work = new double[m];
			bool wantu = true;
			bool wantv = true;

			// Reduce A to bidiagonal form, storing the diagonal elements
			// in s and the super-diagonal elements in e.

			int nct = Math.Min(m - 1, n);
			int nrt = Math.Max(0, Math.Min(n - 2, m));
			for(int k = 0; k < Math.Max(nct, nrt); k++) {
				if(k < nct) {
					// Compute the transformation for the k-th column and
					// place the k-th diagonal in s[k].
					// Compute 2-norm of k-th column without under/overflow.
					s[k] = 0;
					for(int i = k; i < m; i++) {
						s[k] = Maths.Hypot(s[k], A[i][k]);
					}
					if(s[k] != 0.0) {
						if(A[k][k] < 0.0) {
							s[k] = - s[k];
						}
						for(int i = k; i < m; i++) {
							A[i][k] /= s[k];
						}
						A[k][k] += 1.0;
					}
					s[k] = - s[k];
				}
				for(int j = k + 1; j < n; j++) {
					if((k < nct) & (s[k] != 0.0)) {
						// Apply the transformation.

						double t = 0;
						for(int i = k; i < m; i++) {
							t += A[i][k]*A[i][j];
						}
						t = (- t)/A[k][k];
						for(int i = k; i < m; i++) {
							A[i][j] += t*A[i][k];
						}
					}

					// Place the k-th row of A into e for the
					// subsequent calculation of the row transformation.

					e[j] = A[k][j];
				}
				if(wantu & (k < nct)) {
					// Place the transformation in U for subsequent back
					// multiplication.

					for(int i = k; i < m; i++) {
						U[i][k] = A[i][k];
					}
				}
				if(k < nrt) {
					// Compute the k-th row transformation and place the
					// k-th super-diagonal in e[k].
					// Compute 2-norm without under/overflow.
					e[k] = 0;
					for(int i = k + 1; i < n; i++) {
						e[k] = Maths.Hypot(e[k], e[i]);
					}
					if(e[k] != 0.0) {
						if(e[k + 1] < 0.0) {
							e[k] = - e[k];
						}
						for(int i = k + 1; i < n; i++) {
							e[i] /= e[k];
						}
						e[k + 1] += 1.0;
					}
					e[k] = - e[k];
					if((k + 1 < m) & (e[k] != 0.0)) {
						// Apply the transformation.

						for(int i = k + 1; i < m; i++) {
							work[i] = 0.0;
						}
						for(int j = k + 1; j < n; j++) {
							for(int i = k + 1; i < m; i++) {
								work[i] += e[j]*A[i][j];
							}
						}
						for(int j = k + 1; j < n; j++) {
							double t = (- e[j])/e[k + 1];
							for(int i = k + 1; i < m; i++) {
								A[i][j] += t*work[i];
							}
						}
					}
					if(wantv) {
						// Place the transformation in V for subsequent
						// back multiplication.

						for(int i = k + 1; i < n; i++) {
							V[i][k] = e[i];
						}
					}
				}
			}

			// Set up the final bidiagonal matrix or order p.

			int p = Math.Min(n, m + 1);
			if(nct < n) {
				s[nct] = A[nct][nct];
			}
			if(m < p) {
				s[p - 1] = 0.0;
			}
			if(nrt + 1 < p) {
				e[nrt] = A[nrt][p - 1];
			}
			e[p - 1] = 0.0;

			// If required, generate U.

			if(wantu) {
				for(int j = nct; j < nu; j++) {
					for(int i = 0; i < m; i++) {
						U[i][j] = 0.0;
					}
					U[j][j] = 1.0;
				}
				for(int k = nct - 1; k >= 0; k--) {
					if(s[k] != 0.0) {
						for(int j = k + 1; j < nu; j++) {
							double t = 0;
							for(int i = k; i < m; i++) {
								t += U[i][k]*U[i][j];
							}
							t = (- t)/U[k][k];
							for(int i = k; i < m; i++) {
								U[i][j] += t*U[i][k];
							}
						}
						for(int i = k; i < m; i++) {
							U[i][k] = - U[i][k];
						}
						U[k][k] = 1.0 + U[k][k];
						for(int i = 0; i < k - 1; i++) {
							U[i][k] = 0.0;
						}
					} else {
						for(int i = 0; i < m; i++) {
							U[i][k] = 0.0;
						}
						U[k][k] = 1.0;
					}
				}
			}

			// If required, generate V.

			if(wantv) {
				for(int k = n - 1; k >= 0; k--) {
					if((k < nrt) & (e[k] != 0.0)) {
						for(int j = k + 1; j < nu; j++) {
							double t = 0;
							for(int i = k + 1; i < n; i++) {
								t += V[i][k]*V[i][j];
							}
							t = (- t)/V[k + 1][k];
							for(int i = k + 1; i < n; i++) {
								V[i][j] += t*V[i][k];
							}
						}
					}
					for(int i = 0; i < n; i++) {
						V[i][k] = 0.0;
					}
					V[k][k] = 1.0;
				}
			}

			// Main iteration loop for the singular values.

			int pp = p - 1;
			int iter = 0;
			double eps = Math.Pow(2.0, - 52.0);
			while(p > 0) {
				int k, kase;

				// Here is where a test for too many iterations would go.

				// This section of the program inspects for
				// negligible elements in the s and e arrays.  On
				// completion the variables kase and k are set as follows.

				// kase = 1     if s(p) and e[k-1] are negligible and k<p
				// kase = 2     if s(k) is negligible and k<p
				// kase = 3     if e[k-1] is negligible, k<p, and
				//              s(k), ..., s(p) are not negligible (qr step).
				// kase = 4     if e(p-1) is negligible (convergence).

				for(k = p - 2; k >= - 1; k--) {
					if(k == - 1) {
						break;
					}
					if(Math.Abs(e[k]) <= eps*(Math.Abs(s[k]) + Math.Abs(s[k + 1]))) {
						e[k] = 0.0;
						break;
					}
				}
				if(k == p - 2) {
					kase = 4;
				} else {
					int ks;
					for(ks = p - 1; ks >= k; ks--) {
						if(ks == k) {
							break;
						}
						double t = (ks != p ? Math.Abs(e[ks]) : 0.0) + (ks != k + 1 ? Math.Abs(e[ks - 1]) : 0.0);
						if(Math.Abs(s[ks]) <= eps*t) {
							s[ks] = 0.0;
							break;
						}
					}
					if(ks == k) {
						kase = 3;
					} else if(ks == p - 1) {
						kase = 1;
					} else {
						kase = 2;
						k = ks;
					}
				}
				k++;

				// Perform the task indicated by kase.

				switch(kase) {
						// Deflate negligible s(p).
					case 1: {
						double f = e[p - 2];
						e[p - 2] = 0.0;
						for(int j = p - 2; j >= k; j--) {
							double t = Maths.Hypot(s[j], f);
							double cs = s[j]/t;
							double sn = f/t;
							s[j] = t;
							if(j != k) {
								f = (- sn)*e[j - 1];
								e[j - 1] = cs*e[j - 1];
							}
							if(wantv) {
								for(int i = 0; i < n; i++) {
									t = cs*V[i][j] + sn*V[i][p - 1];
									V[i][p - 1] = (- sn)*V[i][j] + cs*V[i][p - 1];
									V[i][j] = t;
								}
							}
						}
					}
						break;

						// Split at negligible s(k).

					case 2: {
						double f = e[k - 1];
						e[k - 1] = 0.0;
						for(int j = k; j < p; j++) {
							double t = Maths.Hypot(s[j], f);
							double cs = s[j]/t;
							double sn = f/t;
							s[j] = t;
							f = (- sn)*e[j];
							e[j] = cs*e[j];
							if(wantu) {
								for(int i = 0; i < m; i++) {
									t = cs*U[i][j] + sn*U[i][k - 1];
									U[i][k - 1] = (- sn)*U[i][j] + cs*U[i][k - 1];
									U[i][j] = t;
								}
							}
						}
					}
						break;

						// Perform one qr step.

					case 3: {
						// Calculate the shift.

						double scale =
							Math.Max(
							         Math.Max(Math.Max(Math.Max(Math.Abs(s[p - 1]), Math.Abs(s[p - 2])), Math.Abs(e[p - 2])), Math.Abs(s[k])),
							         Math.Abs(e[k]));
						double sp = s[p - 1]/scale;
						double spm1 = s[p - 2]/scale;
						double epm1 = e[p - 2]/scale;
						double sk = s[k]/scale;
						double ek = e[k]/scale;
						double b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/2.0;
						double c = (sp*epm1)*(sp*epm1);
						double shift = 0.0;
						if((b != 0.0) | (c != 0.0)) {
							shift = Math.Sqrt(b*b + c);
							if(b < 0.0) {
								shift = - shift;
							}
							shift = c/(b + shift);
						}
						double f = (sk + sp)*(sk - sp) + shift;
						double g = sk*ek;

						// Chase zeros.

						for(int j = k; j < p - 1; j++) {
							double t = Maths.Hypot(f, g);
							double cs = f/t;
							double sn = g/t;
							if(j != k) {
								e[j - 1] = t;
							}
							f = cs*s[j] + sn*e[j];
							e[j] = cs*e[j] - sn*s[j];
							g = sn*s[j + 1];
							s[j + 1] = cs*s[j + 1];
							if(wantv) {
								for(int i = 0; i < n; i++) {
									t = cs*V[i][j] + sn*V[i][j + 1];
									V[i][j + 1] = (- sn)*V[i][j] + cs*V[i][j + 1];
									V[i][j] = t;
								}
							}
							t = Maths.Hypot(f, g);
							cs = f/t;
							sn = g/t;
							s[j] = t;
							f = cs*e[j] + sn*s[j + 1];
							s[j + 1] = (- sn)*e[j] + cs*s[j + 1];
							g = sn*e[j + 1];
							e[j + 1] = cs*e[j + 1];
							if(wantu && (j < m - 1)) {
								for(int i = 0; i < m; i++) {
									t = cs*U[i][j] + sn*U[i][j + 1];
									U[i][j + 1] = (- sn)*U[i][j] + cs*U[i][j + 1];
									U[i][j] = t;
								}
							}
						}
						e[p - 2] = f;
						iter = iter + 1;
					}
						break;

						// Convergence.

					case 4: {
						// Make the singular values positive.

						if(s[k] <= 0.0) {
							s[k] = (s[k] < 0.0 ? - s[k] : 0.0);
							if(wantv) {
								for(int i = 0; i <= pp; i++) {
									V[i][k] = - V[i][k];
								}
							}
						}

						// Order the singular values.

						while(k < pp) {
							if(s[k] >= s[k + 1]) {
								break;
							}
							double t = s[k];
							s[k] = s[k + 1];
							s[k + 1] = t;
							if(wantv && (k < n - 1)) {
								for(int i = 0; i < n; i++) {
									t = V[i][k + 1];
									V[i][k + 1] = V[i][k];
									V[i][k] = t;
								}
							}
							if(wantu && (k < m - 1)) {
								for(int i = 0; i < m; i++) {
									t = U[i][k + 1];
									U[i][k + 1] = U[i][k];
									U[i][k] = t;
								}
							}
							k++;
						}
						iter = 0;
						p--;
					}
						break;
				}
			}
		}

		#endregion	//Constructor

		#region Public Properties

		/// <summary>Return the one-dimensional array of singular values</summary>
		/// <returns>     diagonal of S.
		/// </returns>
		public virtual double[] SingularValues {
			get { return s; }
		}

		/// <summary>Return the diagonal matrix of singular values</summary>
		/// <returns>     S
		/// </returns>
		public virtual Matrix S {
			get {
				var X = new Matrix(n, n);
				var S = X.Array;
				for(int i = 0; i < n; i++) {
					for(int j = 0; j < n; j++) {
						S[i][j] = 0.0;
					}
					S[i][i] = s[i];
				}
				return X;
			}
		}

		#endregion //  Public Properties

		#region	 Public Methods

		/// <summary>Return the left singular vectors</summary>
		/// <returns>     U
		/// </returns>
		public virtual Matrix GetU() {
			return new Matrix(U, m, Math.Min(m + 1, n));
		}

		/// <summary>Return the right singular vectors</summary>
		/// <returns>     V
		/// </returns>
		public virtual Matrix GetV() {
			return new Matrix(V, n, n);
		}

		/// <summary>Two norm</summary>
		/// <returns>     max(S)
		/// </returns>
		public virtual double Norm2() {
			return s[0];
		}

		/// <summary>Two norm condition number</summary>
		/// <returns>     max(S)/min(S)
		/// </returns>
		public virtual double Condition() {
			return s[0]/s[Math.Min(m, n) - 1];
		}

		/// <summary>Effective numerical matrix rank</summary>
		/// <returns>     Number of nonnegligible singular values.
		/// </returns>
		public virtual int Rank() {
			double eps = Math.Pow(2.0, - 52.0);
			double tol = Math.Max(m, n)*s[0]*eps;
			int r = 0;
			for(int i = 0; i < s.Length; i++) {
				if(s[i] > tol) {
					r++;
				}
			}
			return r;
		}

		#endregion   //Public Methods

		// A method called when serializing this class.

		#region ISerializable Members

		void ISerializable.GetObjectData(SerializationInfo info, StreamingContext context) {}

		#endregion
	}
}
